How does the nounはnounでも works?

Question

揚げ物は揚げ物でも天ぷらっていってね

Upon the other character pointing out this is 揚げ物 the character explains this. What kind of nuance does the repeated noun give

Answer 1

It's used to say that something indeed belongs to the general category described by the noun, 揚げ物 in this example, but there is something specific to it, in this case it has a particular name 天ぷら.

[N]は[N]でも[noun-modifier][N]は何？ is a common format for wordplay riddles like these.

Answer 2

でも is often thought of as a single particle, but here it's more easily understood by analyzing it as separate で+も. That is, the て-form of だ, followed by the ordinary も particle.

Taking it step by step:

揚げ物は揚げ物(だ) "agemono is agemono"

揚げ物は揚げ物で "agemono being agemono" ("it being the case that agemono is agemono")

揚げ物は揚げ物でも "even if / despite that agemono is agemono" ("also in the case of agemono being agemono")

So overall, something like "(This) agemono may be agemono, but call it tempura, ok?"

Repetition of this sort is idiomatic enough in Japanese and as far as I'm aware it doesn't add any particular nuance beyond the obvious meaning. The literal translation of one such instance became memetic for English anime fans in the late 00s.